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Research Laboratory 

of the 

Portland Cement Association 



Bulletin 10 



Interpretation of 

Phase Diagrams of 

Ternary Systems 



BY 
L. A. DAHL 



March, 1946 
Chicago 



Reprinted from The Journal of Physical Chemistry 
Vol. 50, No. 3, March. 1946 



Research Laboratory 

of the 

Portland Cement Association 



Bl LLETIN 10 



Interpretation of 

Phase Diagrams of 

Ternary Systems 



in 
I \. DAHL 



\I V, 






Made in United States of America 

Reprinted from The Journal of Physical Chemistry 
Vol. 50, No. 2. March, 1946 



INTERPRETATION OF PHASE DIAGRAMS OF TERNARY SYSTEMS 

L. A. DAHL 

Research Laboratory of the Portland Cement Association, 33 W. Grand Avenue, 

Chicago, Illinois 

Received November 10, 194$ 

Conventional interpretations of phase diagrams of systems of the refractory 
oxides describe the order of appearance and disappearance of solid phases when 
liquids of various compositions are cooled slowly. In applying phase equilibria 
data to problems pertaining to portland cement manufacture, in which raw ma- 
terial mixtures are only partially fused, it has been found advantageous to fol- 
low a somewhat different mode of attack. Methods have been developed for de- 
termining the phases present at equilibrium at any given temperature, and esti- 



PHASE DIAGRAMS OF TERNARY SYSTEMS 97 

mating their proportions, without tracing the course of crystallization from the 
liquid state. In addition, attention has been given to the role' of the liquid 
phase in the processes of fusion and crystallization, as a means of gaining an 
understanding of the mechanism by which the oxides in the raw materials are 
transformed into the various cement compounds. These methods may be found 
useful in other fields, and are consequently presented in this paper in a study of 
hypothetical ternary systems. The paper deals only with condensed systems 
in which the solid phases are practically immiscible. 

IDEAL CONDITIONS OF CRYSTALLIZATION 

In descriptions of the course of crystallization it is customary to assume an 
ideal condition, in which cooling is so slow that at each temperature a state of 
equilibrium is attained before a further reduction in temperature takes place. 
This condition may be described as a continuous attainment of equilibrium 
during the process of cooling. When there is a continuous attainment of equilib- 
rium, the phase diagram supplies all of the information necessary to predict 
the phases present at any stage in the process, and to estimate their proportions, 
by a process of deduction involving the mathematical properties of the triangular 
diagram. 

With a slow attainment of equilibrium the process of cooling must be extremely 
slow if a continuous attainment of equilibrium is to be secured. The conclusions 
in regard to the course of crystallization, obtained by assuming a continuous 
attainment of equilibrium, consequently apply only to cases in which the process 
of cooling is so slow that this condition is secured. In an industrial process the 
condition must be known to exist before it can be assumed that the predicted 
changes in character and proportions of phases actually occur. It is not enough 
to know that a time element is involved in the process. It must be known that 
the rate of temperature change in the process is sufficiently slow to secure the 
continuous attainment of equilibrium. This cannot be known from the phase 
diagram, but can only be known by observations in addition to those represented 
in the diagram. 

The purpose of this paper is to present methods of interpreting the phase 
diagram with reference to industrial processes. It will consequently not be 
confined to a study of changes occurring with a continuous attainment of equi- 
librium, but will deal also with the changes which may occur if temperature 
change is too rapid to permit such a condition to exist. Because of the fact 
that most discussions of the course of crystallization assume the continuous 
attainment of equilibrium, it will be necessary to depart from conventional 
methods of presenting the subject, and to adopt an entirely new method of 
treatment. 

CONSIDERATIONS IN A CONDENSED SYSTEM 

A condensed system is one in which a great increase in pressure is required to 
produce a small change in the temperature required for a given state of equilib- 
rium. Phase equilibria in such systems may be investigated in open vessels 



98 L. A. DAHL 






at atmospheric pressure, and yet yield data practically identical with those 
which would be obtained if the mixtures employed in the investigation were 
subjected only to the pressure of their own vapor. Since large changes in 
pressure are required to produce only a small change in the temperature required 
for a given state of equilibrium, the interpretation of the phase diagram of a 
condensed system does not involve consideration of pressure as a variable. In 
stating conditions imposed upon a system, temperature on]}- will be mentioned. 

'When the composition and temperature of a condensed system are given, the 
character and proportions of phases, that is, the phase composition at equilib- 
rium, is defined, except at the temperature at an invariant point, 1 and may be 
estimated from the phase diagram. Although the complete phase composition 
includes the vapor phase, the amount of material involved in the vapor phase is 
so small that it may be ignored. The vapor phase will consequently be ignored 
in discussing phase compositions, or changes in phase composition resulting 
from decrease in temperature (the process of crystallization) or from increase 
in temperature (the process of fusion). 

According to modern theories of liquid structure, molecules may be associated 
in large aggregates, some of them in a relatively orderly arrangement. However, 
phase equilibria relations in a condensed ternary system are compatible with 
the simple assumption that liquids in the system are composed entirely of un- 
associated molecules of various kinds. Since this assumption is sufficient for 
the purpose at hand and helps to simplify treatment of the subject, it is adopted 
in the study which folio v 

HYPOTHETICAL SYSTEM INVOLVING THREE SOLID PHASES 

Let us consider first a simple ternary system involving three solid phases, 

A. B, and ('. All mixtures of A, £, and C may be represented in a triangular 

diagram in the customary manner. If mixtures of a variety of compositions, 

:<e entire diagram, are melted, and the temperature of complete 

ion is determined in each case, the data obtained establish isotherms in the 
diagram, as shown in figure L 2 Passing from A to B, that is, beginning with 
and considering g e additions of J5, it is seen from the diagram that the 

temperature of complete fusion drops from 117 c at .4 to 75° at H r and then 
inc " 12 ''" at B - The point // represents a eutectic composition, with a 

lower temperature of complete fusion than either .4 or B. Similarly, K is a 
eutectic mixture trf .1 and C, and is a eutectic mixture of B and C. The point 
E : - a eutectic mixture of the three components .4, B y and C, with a 

lower temperature of complete fusion than any other mixture in the system. 

The 50° and 60° isotherms are not continuous curves, but change abruj 
' m direction - a closed figure. If isotherms are drawn to represent te- 

mperatures, similar abrupt changes in direction will appear. The 

iriaotpointth, imlibritm 

not by composition 

I merely f or eonvenii 



m 

Th 

Hi 



PHASE DIAGRAMS OF TERNARY SYSTEMS 



99 



curves EK, EH, and EG are the loci of the points at which such changes in 
direction occur. These curves divide the triangle into three regions. When 
any mixture within the region EKCG is maintained at its temperature of com- 
plete fusion, it is capable of existing in equilibrium with solid phase C. Above 
that temperature it is liquid. Just below that temperature, if no supercooling 
occurs, the solid phase C separates out from the liquid. It is the first crystalline 
phase to appear in the normal process of crystallization, and is called the pri- 
mary phase for that region. Similarly, solid phase A is the primary phase for 
the region AKEH, and -olid phase B is the primary phase for the region HEGB. 




Fig. 1. Typical phase diagram of t< rnary system involving three molecular species. 



The three regions are termed primary-phase regions. The letters A, B 7 and C 
in the three regions designate the primary phases. 

When a system is in a state of equilibrium, the escaping tendency of each 
molecular species involved is constant throughout the system. Consider a 
liquid represented by a point on the line RS } which is the branch of the 60° 
isotherm in the C primary-phase region. When the liquid is in equilibrium with 
solid C at 60°, the escaping tendency of C from the liquid is equal to that of 
crystalline C. It is this balance between escaping tendencies which makes it 
possible for the two phases, solid C and liquid, to exist together without change. 
The term equilibrium implies a balance between forces or reactions, in this 
case escaping tendencies. 

From the foregoing description of the conditions existing in a state of equilib- 



100 



L. A. DAHL 



rium, it is evident that at 60° the escaping tendency of the component C 
from all liquids on RS is the same. While the numerical value of this escaping 
tendency may not be known, it is known to be identical with that of solid C at 
60°. The line RS consequently has a significance in addition to that of rep- 
resenting the locus of compositions with a temperature of complete fusion of 
60°. It refers also to a particular solid phase, C, and may consequently be 
referred to as the 60° C isotherm. All other isotherms in the C primary-phase 
region may thus be described as C isotherms. Similarly, all isotherms in the 
.4 primary-phase region are A isotherms, and all in the B primary-phase region 
are B isotherms. 

The isotherms being considered with reference to temperatures of complete 
fusion, the figure RST may be regarded as a single isotherm, changing abruptly 
in direction at R, S, and T. From the standpoint of their significance with re- 
spect to escaping tendencies, the three branches must be regarded as separate 
isotherms, the 60° .4 isotherm RT, the 60° B isotherm ST, and the 60° C 
isotherm RS. 

Significance of the isotherm — supercooling 
In ternary phase diagrams the separate isotherms meeting at the boundaries 
of two primary-phase regions are usually drawn only to those boundaries, giving 
the appearance of being single isotherms changing abruptly in direction. From 
the standpoint of their true significance, with reference to escaping tendencies, 
they may be considered as extending beyond the boundaries. For example, the 
50° A isotherm in figure 1 is extended by a broken line to the point P. If a 
liquid of composition P can be supercooled to 50° (through failure of solid 
C to appear), the escaping tendency of component A from the liquid will then 
be equal to that of pure solid .1 at that temperature. The liquid P will be ca- 
pable of existing in equilibrium with solid A at 50°, and such a state of met- 
astable equilibrium may be maintained indefinitely if solid C does not appear. 
The point 2\ figure 1, is on both the A and B 60° isotherms. At 60°, 
then, the escaping tendency of component A from liquid T is equal to the es- 
caping tendency of solid A at 60°, and the liquid may exist in equilibrium with 
solid .4. Since it is also on the 60° B isotherm, liquid T may exist in equilib- 
rium with solid B. That is, as a result of the fact that T is on both the A and 
B isotherms for 60 c , the liquid is capable of existing in equilibrium with two 
solid phases, .4 and B. From this it is seen that all liquids on EH may exist 
in equilibrium with two solid phases, ,4 and J3, or with either phase alone, at the 
indicated temperatures. Similarly, liquids on EK may exist in equilibrium with 
the solid phases ,1 and r ', and liquids on EG may exist in equilibrium with B and 
( , at the indicated temperatui 

( The triangular phase diagram of a ternary system, such as that shown in 
figure 1, is the projection of a .pace model, in which the base is a triangular 
diagram and temperature is represented by distances in a direction perpendic- 
ular to the base. In this .pace model the primary-phase regions are curved 
surfaces meeting in "valleys" which, when projected on the base, are represented 



PHASE DIAGRAMS OF TERNARY SYSTEMS 101 

by boundaries between primary-phase regions. A horizontal section through 
the space model at a distance from the base corresponding to a particular tem- 
perature will cut the curved surfaces along curved lines which are isotherms 
The isotherms in the phase diagram are projections of these lines on the base. 
Viewing figure 1 in this manner we can see that in the space model there are 
three surfaces sloping down from the vertices to EK, EH, and EG y and that 
they intersect at E, the lowest point. The surface AKEH, representing con- 
ditions under which liquids may exist in equilibrium with solid A, may be 
termed the A surface. Similarly, the surface HEGB is the B surface, and EKCO 
the C surface. 

Attention has been called to the fact that the isotherms in one region may be 
extended into another region, as shown in extending the 50° A isotherm to P. 
This is equivalent to an extension of one surface beyond the curve of intersection 
with another surface (the curves EK } EH, and EG). In that case the A surface, 
extended, passes under the B and C surfaces, etc. Each surface shown in the 
diagram is the upper surface in the region indicated. 

Other molecular species, that is, various combinations of .4, B, and C, may be 
present in the liquid. However, since these molecular species are not represented 
by surfaces above those of A, J5, and C, they may be ignored in considering 
solid-liquid equilibria. As A, J3, and C are removed from the liquid in the proc- 
ess of crystallization, these other molecular species dissociate to release more A, 
B, and C. There is consequently no residue of other substances when crystalli- 
zation is complete. 

If the surfaces are considered as extending beyond the limits shown, it is 
evident that there will be a 40° A isotherm below the B and C surfaces. If 
42" J isotherms are drawn for the three surfaces, they will intersect at E. Since 
a liquid of composition E at 42° is on the A, B 7 and C 42° isotherms, it is 
capable of existing in equilibrium with the three solid phases A, B, and I 

Tracing the course of crystallization 

The phase diagram designates the phases which may exist in equilibrium in a 
heterogeneous system at a given temperature, but does not give directly any 
other information concerning the phases present, or their proportions, in the 
system. That is, one cannot locate in the phase diagram a point representing 
the composition of a particular system and then determine the phase compo- 
sition by mere inspection. However, the phases which may exist in equilibrium 
are given in the diagram and only certain combinations of these phases are 
mathematically capable of forming any given composition. It is consequently 
possible to combine the information from the phase diagram with the mathemat- 
ical possibilities concerning the combinations of phases which may form the 
composition of a mixture, and thus determine the phase composition. In this 
study an effort will be made to distinguish between the information obtained 
from the phase diagram and that obtained solely from mathematical consider- 
ations. 

For our first problem, let us assume that a mixture of composition M (figure 



102 



L. A. DAHL 



1) is maintained at 70 : . The problem is to determine the phase composition 
when equilibrium is attained. The position of .1/ with reference to the isotherms 
indicates that its temperature of complete fusion is greater than 70° (about 
75°). Hence it is known that the mixture M will be partly solid at 70°. 
Since the 70* ( i-otherm is entirely within the C primary-phase region, in 
which M lies, it is known that the mixture .V will be composed of solid C and 
liquid when in equilibrium at 70 c . Only liquids on the 70° C isotherm mav 
exist in eqmhbrium with solid C at that temperature. From the phase diagram, 
then, it is known that the phases are solid C and a liquid which must be on the 
70° C isotherm. This is all that is known from the conditions of phase equi- 
libria represented in the phase diagram. To determine the particular liquid 
on the 70 c C isotherm which will be present, a knowledge of the mathematical 
properties of the triangular diagram is requin 

When two substances represented I in a triangular diagram are mixed, 

all compositions which may be formed by adjusting their proportions are lo- 
cated on the straight line joining the points. For example, if N is mixed with C 
(figure 1), all of the possible compositions are on the line XC. Although N 
is on the 70 c C isotherm, and therefore fulfills the condition set by the require- 
ment of equilibrium between solid C and liquid, it is known from mathematical 
considerations that it cannot be the liquid present in M when a state of equilib- 
rium is attained at 70 : . since M is not on the line AT. The only liquid which 
fulfills both the phase equilibrium requirement he liquid must be on the 

>therm) and the mathematical requirement that the point M must be 
on the line joining the two phases) is the point L. It Ls consequently known 
that when equilibrium is attained at 70 c the mixture M is composed of solid C 
and liquid L. 

The phases have been determined, and it is now necessary to determine their 
proportions. If the length of the line LC is taken as unity, the segment ML 

rthest in a I he fractional proportion of solid C. Similarly, the segment 

I (farthest from L) is the fractional proportion of liquid L. By measurement 
it is found that the fractional proportion of solid C is 0.13 (or 13 per cent); the 
fractional proportion of liquid L is 0.87 (or 87 per c* ■: 

the mixta -,.oled to 56° and maintained at that temperature until 

equilibrium is attained, the composition of the liquid pha^e must then be on the 
55 c C isotherm, about halfway between the 50° and <;<) isotherms and also 
on an extension of the line CM, that is, at U The system is then composed 
of solid C and liquid L : . The effect of reducing the C content of the liquid, in 
this case by crystallisation, is to change ife composition in a direction awav 
from the point C. This may be stated briefly, speaking with reference to the 
position of the liquid in the diagram, by saying that the removal of C from 
the liquid forces the liquid directly away from I 

Xow if the mixture is cooled to 50°, with no other phenomenon but the 
separation ot solid liquid is forced to L,, on an extension of the 50° 

C isotherm. However, the liquid Is now in the B primarv-phase region If B 
separ the liquid is forced directly away from B, that is, toward a point 

on EG between L, and U The combined effect of the tendency to change in a 






PHASE DIAGRAMS OF TERNARY SYSTEMS 103 

direction away from C, when in the C primary-phase region, and away from B, 
when in the B primary-phase region, is to cause the liquid composition to follow 
the boundary curve EG. If the solid phase B fails to appear, the liquid will 
have the composition L 2 when equilibrium is attained. It is then in a state of 
metastable equilibrium, since the state of equilibrium may be disturbed in 
various ways, causing the separation of B. If crystallization of B is sluggish, 
the composition of the liquid will follow a path in the B primary-phase region 
between the extended 50° C isotherm and the boundary curve EG, finally ar- 
riving at L 3 when equilibrium is attained. If crystallization of B is rapid, the 
composition of the liquid will follow the boundary curve EG from Li to L 3 . 
When equilibrium is attained at 50°, the mixture M will be composed of solid 
B, solid C, and liquid L 3 . 

If the mixture is now cooled to 42° the liquid will follow the boundary curve 
EG to E, with further separation of solid B and C. When the liquid arrives at E 
the mixture is composed of solid B, solid C, and liquid E. At this point the 
liquid is capable of existing in equilibrium with the three solid phases A, B f and C. 
Upon further removal of heat, solid A, B y and C separate out, until the last 
trace of liquid disappears. This occurs without change of temperature, since 
the heat which is withdrawn is the latent heat released in the process of crys- 
tallization. When crystallization has been completed at 42°, further removal 
of heat results in a decrease in temperature. 

It was mentioned earlier that during the crystallization of mixture M the 
liquid is forced to follow the boundary curve EG by the crystallization of B. If 
solid B fails to separate out, the liquid will follow the extension of the line M-Lx 
into the B primary-phase regions. A similar situation arises after the liquid 
arrives at E along the boundary curve EG. If C appears, the liquid remains at 
E until crystallization is complete. If solid C fails to appear, the liquid will 
follow an extension of the boundary curve EG into the A primary-phase region. 

Determination of phase composition without tracing the course of crystallization 
The phases present in the mixture M when a state of equilibrium is attained at 
50° have been found by following the course of crystallization from higher 
temperatures, the customary procedure in studies of the course of crystallization. 
It must not be supposed, however, that the phases present at equilibrium can be 
determined only by starting with the fused mixture, and tracing the appearance 
and disappearance of solid phases as the mixture is cooled. The equilibrium 
state at 50° is the same whether the mixture is first fused and then cooled, or 
is heated from a lower temperature to 50°. This should be clearly understood, 
since in many cases as, for example, in the production of portland cement clinker, 
the original solid materials are only partially fused. In studying such processes 
it is convenient to be able to determine the phases present without tracing the 
course of crystallization from complete fusion to the required temperature. On 
that account the phases present in the mixture M at equilibrium at 50° will be 
determined again, this time without tracing the path followed by the liquid 
composition (the crystallization curve) from M to L h and then to L 3 . 

First of all, it is known that the mixture is not composed of only two phases, 



104 L. A. DAHL 

solid and liquid, since no point on the 50° C isotherm is in line with C and If. 
If three phases are present the}' must be the phases designated at a 50° point 
on a boundary curve, such as La, L 4 , or L 6 . Any mixture composed of three 
phases must be within the triangle formed by joining the points representing the 
compositions of the phases by straight lines. For example, since L 4 is on the 
boundary between the A and C primary-phase regions, the only mixtures in 
which L 4 may be present at equilibrium at 50° are those within the triangle 
formed by drawing straight lines from A to C, from C to L 4 , and from L 4 to A. 
By mere inspection it is seen that M does not lie in the triangle, and it is known, 
therefore, that the liquid L 4 is not present in M under the required conditions. 
Similarly, if L 3 is the liquid phase the point M must be within the triangle formed 
by joining L 3 , B, and C by straight lines, which it is. These three phases, solid 
B, solid C, and liquid L 3 , are therefore the phases present. 

With a simple phase diagram, such as figure 1, and with experience in tracing 
the course of crystallization, one would naturally test L 3 first as the possible 
liquid phase, with no need of inspecting a number of such points before finding 
L 3 to be the liquid phase. However, without such experience, or in more com- 
plicated situations, the method just described may be used. It has been given 
merely to demonstrate the fact that the phases present at equilibrium may be 
determined without tracing the course of crystallization. 

The phases in M at equilibrium at 50° have been determined by two methods, 
but their proportions have not been estimated. In discussions of the course of 
crystallization it is customary to estimate the relative proportions of liquid and 
solid, and then to estimate the relative proportions of the two solid phases in the 
solid portion. This is not entirely satisfactory, since it is possible for a solid 
phase to decrease in amount and yet form an increasing proportion of the total 
solids, or vice versa. By following this procedure, the direction of change in the 
amount of a particular solid phase during crystallization or fusion can be deter- 
mined, but only in an indirect fashion. In this study a direct method will be 
employed, in which the proportions of the liquid and the two solid phases are 
estimated simultaneously. 

The points representing the mixture .1/ and the liquid phase L 3 have been trans- 
ferred from figure 1 to figure 2. The triangle B-L 5 -( ' includes the composition of 
all mixtures which are composed of solid B, solid (\ and liquid L 3 at equilibrium 
at 50°. Broken lines are drawn from M, parallel to two sides, dividing the 
third side into segments proportional to the amounts of the three phases. If the 
length of BC is taken as unity, the segment VW is the fractional proportion of 
liquid L 3 (at the vertex opposite BC), the segment CV (farthest from B) is the 
fractional proportion of B, and the segment B\V (farthest from C) is the frac- 
tional proportion of C, in the mixture M. Any side of the triangle may be sub- 
divided by this procedure for estimating phase composition. 

The examples given have illustrated methods of determining phase composition 
when the liquid phase is located in a primary-phase region or on a boundary 
between primary-phase regions. In each ease the phase composition of a system 
in a state of equilibrium is completely determined by the composition and tern- 



PHASE DIAGRAMS OF TERNARY SYSTEMS 



105 



perature. When the liquid phase is at an invariant point, however, the phase 
composition is not completely determined by composition and temperature. On 
that account in considering figure 1, it is necessary to study separately the phase 
composition when a state of equilibrium is attained at 42°, the temperature 
designated at the point E. 

The liquid phase E is capable of existing in equilibrium with the three solid 
phases A,Bj and C. The presence of any one of the four phases is not necessary 
for the existence of a state of equilibrium between the other three. That is, by 




Fig. 2. Diagram illustrating graphic method of estimating phase composition with three 
phases present at equilibrium. 



successively eliminating one phase at a time from the four phases, four systems 
of three phases each are obtained, each of these representing equilibrium states 
at 42°, as follows: 

I. Solid B, solid C, liquid E 

II. Solid A, solid C, liquid E 

III. Solid A, solid B, liquid E 

IV. Solid .4, solid 5, solid C 

If system I is considered, the two solid phases are the phases capable of existing 
in equilibrium with liquids on the boundary curve EG at designated temperatures. 
The system consequently refers to E as a point on the line EG, which is the curve 



106 



L. A. DAHL 



followed by the liquid phase during slow cooling of certain mixtures originally in 
a state of equilibrium above 42°, as was illustrated by a study of the mixture M. 
When such a mixture is slowly cooled to 42°, so slowly that a practically con- 
tinuous state of equilibrium is attained, it will consist of solid B, solid C, and 
liquid E. The compositions which may attain such a state of equilibrium at 
42° are included in the triangle formed by joining the points B, C, and E by 
straight lines, as shown in figure 3, to which the point E in figure 1 has been trans- 
ferred. This triangle therefore represents system I. Similarly, system II is 
represented by triangle ACE, and system III by triangle ABE, in figure 3. 




Fig. 3. Systems with two solid phase? and a eutectic liquid E. Maximum heat content 
at eutectic temperature dmum proportion of liquid at eutectic temperature). 

It will be observed that each triangle in figure 3 represents a system composed 
of three phases, one of them liquid, at equilibrium at 42\ The phase com- 
position of any mixture under these conditions may be determined. For ex- 
ample, the proportions of solid B, liquid E, and solid C in the mixture M, which 
is the same as M in figures 1 and 2. are indicated by the segments CV h V t W u 
and WiB, respectively. Phase compositions may also be represented by draw- 
ing lines parallel to the three Bides, as in system III, figure 3, in the same manner 

is customary for the equilateral triangular diagram. 

and minimum heal content 

When heat is withdrawn from any mixture in the state of equilibrium repre- 
sented by the three systems in figure 3, a fourth phase appears, if there is no 



PHASE DIAGRAMS OF TERNARY SYSTEMS 107 

supercooling. That is, liquid E and solids A, B, and C are then present. As 
heat is withdrawn the amount of liquid decreases without change of temperature, 
until finally only solid A, solid B, and solid C are present. The systems I, II, 
and III are therefore systems of maximum heat content at 42°, and refer to the 
state of equilibrium at 42° in which a maximum of liquid E is present. System 
IV, which represents the state of equilibrium at 42° when sufficient heat is re- 
moved to cause all liquid to disappear, is the system of minimum heat content at 
42°. Since the phases then are the solids A, B, and C, the entire triangle ABC 
is required to represent system IV, as shown in figure 4. 




Fig 4. System with three solid phases at eutectic temperature. (Final products of 
crystallization. Minimum heat content at eutectic temperature. 

The phase composition when crystallization is completed at 42° may be 
estimated by either method described in connection with figure 3. Both methods 
are represented in figure 4. 

In showing only a portion of the phase diagram it is sometimes convenient to 
designate the 10 per cent intervals along the sides of the diagram, as shown along 
part of the base AB in figure 4. These points are represented by short segments 
of the longer lines in the figure, drawn parallel to the sides of the triangle. 

From the foregoing study it should be clear that the states of equilibrium rep- 
resented by an invariant point include a range of possible phase compositions, 
from maximum to minimum heat content, at a single temperature. Between 
the maximum and minimum heat content four phases are present. The phase 



108 L. A. DAHL 

composition at equilibrium is consequently defined not by composition and 
temperature but by composition and heat content. 

For condensed systems it may be stated in general that the phase composition 
of a system is defined b}- composition and heat content. The phase composition 
is defined by composition and temperature in all cases in which change in heat 
content is accompanied by a change in temperature. These latter cases include 
all states of equilibrium represented in the phase diagram except those in which 
the liquid is at an invariant point. 

In setting up the diagram in figure 1 it was assumed that only three solid phases 
were involved in the crystallization of liquids composed of the components .4., B, 
and C. It was recognized that other molecular species, that is, various combina- 
tions of A, B, and C, might be present in the liquid, but these do not appear as 
solid phases. It will now be assumed that one of these molecular species, the 
compound AC> appears in the crystallization of liquids composed of A $ B, and C. 
The surface representing the relation of escaping tendencies of AC in the liquid 
to that of solid AC must then be the upper surface in some region in the phase 
diagram. 

If there were no reactions m which A and C combine to form AC, and AC 
dissociates into A and C, the system A-B-l -A ( would have to be treated as a 
quaternary system, requiring a space model to represent phase equilibria rela- 
tions. On the other hand, since such reactions do occur in the liquid phase, the 
system A-B-C is a ternary system, with a solid phase AC as one of the products of 
crystallization. Any state of equilibrium involves a state of homogeneous equi- 
librium in the liquid phase and a state of heterogenous equilibrium with reference 
to the phases present. Under these conditions the disturbance of the equilibrium 
state in the liquid caused by the removal of A , C, or AC in the process of crystalli- 
zation results in reactions within the liquid in a direction which will supply more 
molecules of the substance separating out. These reactions may be successive 
reactions involving molecular species other than those which have been con- 
sidered. The net result, however, may be represented by the reversible reaction, 

A + C^±A( 

which proceeds toward the right if ACk removed from the liquid, and to the left 
if A or C is removed. For the purpose of this study it is possible to ignore the 
presence of molecular species other than those represented by primary-phase 
regions, and to refer to the liquid as being composed of .4, B, C, and AC. 

Introduction of Ac equilibrium wurfact 

It will be assumed that the A, B t and C surfaces in figure 1 also exist in the e 
tem now under consideration in which AC is a product of crystallization. These 
been transferred to figure 5, and AC isotherms added to represent 
contours of the AC surface. Dotted lines in these figures represent loci of inter- 
sections of the AC isotherms with corresponding isotherms on the A, B y and C 
surfaces. These lines bound a region AV£V£ r /v 2 in which the AC surface is 
above the .4, B, and C surface,. This Is the primary-phase region for AC, as 
shown in figure 6. 






PHASE DIAGRAMS OF TERNARY SYSTEMS 



109 




Fig. 5. Diagram illustrating effect of adding a fourth molecular species to system origi- 
nally assumed to involve three molecular species. 




Fig. 6. Phase diagram resulting from hypothesis assumed in figure 5 



The A, B y and AC surfaces slope downward toward the point E 2} where they 
intersect,' The point E 2 is therefore the eutectic for these three solid phases. A 
liquid of composition E 2 may exist in equilibrium with solid A, B, and AC at the 
temperature indicated (about 58°), Similarly, E l is the eutectic for the three 
solid phases B, C, and AC. The eutectic E for the solid phases A, B, and C 
(figure 1) does not appear, but is still present below the AC surface. It is signifi- 
cant under the present assumption only in case of supercooling in which the solid 
phase AC fails to appear. 

Since solids B, C, and AC may exist in equilibrium with liquid Ei at 56°, it is 
known that any two or any three of these phases may exist in equilibrium at 56°. 
Thus it is known that the three solid phases B, C, and AC may exist in equilib- 
rium. The triangle B-C-AC includes all possible mixtures of the three solid 
phases. The solid phases at the vertices are the solid phases present when com- 
positions within the triangle are completely crystallized. Similarly, the solid 
phases at the vertices of the triangle A-B-AC are the final products of crystalliza- 
tion of all compositions within this triangle. 

Construction of triangles representing final prod ads of crystallization 
The broken line AC-B divides the diagram into two triangles, each represent- 
ing final products of crystallization of all compositions within the triangle. In 
the interpretation of phase diagrams of this type, the lines dividing the phase 
diagram into triangles representing final products of crystallization are required. 
A simple method may be given for determining which solid phases are to be joined 
by straight lines for that purpose. 3 For each pair of solid phases which appear as 
primary phases in adjacent regions a straight line is drawn between the points 
representing their respective compositions. For example, the AC and B primary- 
phase regions are adjacent, A straight line is therefore drawn from AC to B. 
Similarly, since the A and AC primary-phase regions adjoin, a line must be drawn 
from A to AC. This line is already dra* n, since it is a segment of the side A-C. 
In figure 6 there is only one line, AC-B, which needs to be added. 

Instead of merely presenting figure 6 as a type of ternary phase diagram which 
might be encountered, it has been constructed on the assumption that AC is one 
of the molecular species in liquids in the system A-B-(\ with its equilibrium 
surface above those of A, B y and C in a particular region. This mode of treat- 
ment has been adopted to form a basis for considering the role of the liquid phase 
during fusion and crystallization of mixtures in the system. 

a' reactions during fusion and crystallization 

In figure 6 four molecular species A, #, ( \ and AC are involved. The propor- 
tion- of solid phases and liquid at a eutectic temperature may be found in the same 
manner as in the foregoing study of figure 1 . Let us suppose that X is a mixture 
of the solid phases A % B, and C. Upon raising the temperature to 50° a liquid 
of <-< imposition E\ will form . When liquid Ei is in a state of homogeneous equilib- 

ig method may be applied to any cond stem in which the solid \ 

ly immiscible. 






PHASE DIAGRAMS OF TERNARY SYSTEMS 111 

rium at 56°, it contains definite proportions of A, B, C, and AC. Since it is 
formed at first from A , B, and C only, it can only attain this state by the reaction, 
A + C — ► AC, the direction of the reaction being determined by the lack of AC. 
Since Ej is above the A surface, A must tend to go into solution. As A, B, and C 
dissolve, the tendency is to form a liquid in which the proportions of A and C are 
such as to force the reaction A + C ^± AC to the right. When sufficient AC is 
formed to cause the liquid to be incapable of existing in equilibrium with solid 
AC 7 then solid AC separates out, thus also assisting in forcing the reaction to the 
right. The tendency, then, is to dissolve solid A and C and form solid AC. 
From the fact that X is in the triangle AC-Ei-B it is known that this process 



Fig. 7. Estimation of phase composition of mixture X (in figure 6) at equilibrium with 
maximum heat content at 56°. 

continues until no solid A and C remain. When equilibrium is attained at 56° 
with maximum liquid (maximum heat content at 56°) the mixture X is com- 
posed of solid AC, solid B y and liquid E ly the latter, however, being composed of 
the four molecular species A, B r C, and AC. The proportions of the three phases 
may be determined graphically, as shown in figure 7 

Starting with composition X in a state of equilibrium at 56° with a maxi- 
mum of liquid E Y present, let us now consider the changes occurring as heat is 
withdrawn from the system. Since E t is at the intersections of the AC, B, and C 
primary-phase regions, it is known that solid AC, B, and C will separate out 
from the liquid as heat is withdrawn, if no supercooling occurs, but that any A 
which is present in the liquid will remain in solution. As AC, B, and C separate 



\\0 L. A. DAHL 



STEM INVOLVING INVARIANT POINTS WHICH ARE NOT EUTECTICS 

Figure 8 is identical with figure 6, but isotherms representing contours of an 
of an AjC surface have been drawn with broken lines, this surface representing 
the conditions under which a solid phase AjC may exist in equilibrium with 
liquids in the system. It Is assumed that there are reversible reactions involving 

•It ii I in this statement that the components A, B, and C are compounds. 






out, the tendency is to increase the concentration of A in the liquid. The in- 
crease in concentration of A caused by the removal of the other three constituents 
of the liquid cannot progress far, -nice any increase in A forces the reaction A + 
C ^ AC (in the liquid) to the right. As crystallization proceeds, this reaction in 
the liquid continues until, when the last drop of liquid disappears, no A remains. 
The mixture then consists of solid AC, B, and C. 

Let us consider again the mixture X when equilibrium is attained at 56° 
with a maximum of liquid E x present. It is then composed of solid AC, solid B, 
and liquid E t . If the composition of X is to be expressed in terms of compounds 
present, rather than phases, percentages of at least four compounds would have 
to be given, since the liquid E 1 contains the four molecular species A, B, C, and 

* For some purposes the composition of X under the condition described 
may be expressed in terms of A. B, and C (the components of the entire system) 
or it may be expressed in terms of AC, B, and C (the final products of crystalliza- 
tion). Although in the latter case composition is expressed in terms of com- 
pounds, it does not represent the actual compound composition of the mixture, 
but merely a mathematical transformation of composition from one system of 
components to another. It is only when crystallization is complete that the 
actual compound composition, or chemical constitution of the mixture, is defined 
by expressing composition in terras of percentages of AC, B, and C, since it Is only 
then fchat the composition expressed in these terms defines the phases and their 
proportions. 

From the example just given it is evident that the process of crystallization in a 
system of the type shown in figure G, in which there are more primary-phase 
regions than the number of compom nts 3 is not a purely physical process. Each 

te of equilibrium represented is a state of balance between escaping tendencies 
of molecular species in the separate phases, as in the simple system considered in 
figure 1. In addition, however, there is a state of homogeneous equilibrium in 
the liquid phase, so that the process of crystallization and the reverse process of 
fusion involve chemical change. 

The original mixture X was composed of ^olid A. B, and C. Upon being par- 
tially fused and again crystallized, it Is composed of the solid phases AC, B, and 
C, not the original solid phases. In this process chemical reactions in the liquid 
phase play a prominent part, as has been shown. It is this function of the liquid 
phase which maker- it possible, and also useful, to consider the individual systems 
A-B-AC and B-C-AC as being part- of a more complete ternary system, the 

-tern A-B-C. 









OF TERNARY SY8 I 



113 




Fig 8 I 
involve four molecul • 




I 



114 L. A, DAHL 

the four molecular species A, AC, A n C, and C, causing their concentrations to 
tend to change in such a way as to attain a state of homogeneous equilibrium. 
These reactions may be expressed in various wa} r s, for example: 

A + C ^ AC 
A +C^± A n C 

However these reactions may be expressed, there is for each composition and 
temperature a definite state of homogeneous equilibrium in the liquid, in which 
the concentrations of A, B, C, AC, and A n C are fixed. The surfaces represent the 
conditions under which any liquid in a state of homogeneous equilibrium may 
exist in equilibrium with the respective solid phases. 

The points at which the A n C isotherms meet the corresponding isotherms in 
other regions are joined by dotted lines, forming a closed figure. In the region 
defined by this figure the A n C surface is the upper surface. The region is there- 
fore the A n C primary-phase region. The A n C isotherms in the region may there- 
fore be drawn with solid lines, and those extending below other surfaces may be 
omitted, as in figure 9. 

Since the primary-phase regions for A n C and B are adjacent, a line is drawn 
between the points representing the compositions of the solid phases A n C and B. 
Similarly a line is drawn from AC to B. The triangle ABC is thus divided into 
triangles representing the final products of crystallization. 

When any mixture in the triangle AC-B-C is cooled slowly to permit the con- 
tinuous attainment of equilibrium, a state of equilibrium is finally attained, at 
56°, in which the liquid E 1 and the solid phases AC, B, and C are present. As 
crystallization proceeds the amounts of the three solid phases increase, until when 
crystallization is completed only the three solid phases are present, This may be 
described briefly by stating that all mixtures in the triangle AC-B-C complete 
their crystallization at E x . In such a statement it is not meant that the composi- 
tion of the mixture is then E u but that E l is the composition of the liquid at the 
last stage of the crystallization process. 

Any mixture in a triangle representing final products of crystallization com- 
pletes its crystallization at the point at which the three phases at its vertices may 
exist in equilibrium with liquid. Mixtures in the triangle A n C-B-AC therefore 
complete their crystallization at I v The point I I is out he triangle. It 

is consequently not classed as a eutectic, but as an "invariant point not a eutec- 

". Similarly, all mixtures in the triangle A-A n C-B complete their crystalliza- 
tion at I 2 , which is also an invariant point, but not a eutectic. 

I 'omparison of eutectic and rum-i atretic invariant points 

In the phase diagram of the type shown in figure 1 all mixtures in the triangle 
A-B-C complete their crystallization at 42°, with a liquid phase E present until 

llizatioo is complete. The liquid E is a eutectic, since it is within the tri- 
angle. The direction of falling temperatures along the boundaries between pri- 
■ --phase regions is toward the eutectic point, that is, from G to E, H to £\ 
and K to E. At the eutectic temperature there are three systems of three pi 












PHASE DIAGRAMS OF TERNARY SYSTEMS 115 

each, of maximum heat content, shown in figure 3, and one system of minimum 
heat content, shown in figure 4, the latter involving only solid phases, the final 
products of crystallization. Each of the triangles in figure 6, A-B-AC and AC- 
B~C y may be given similar treatment. 

At an invariant point which is not a eutectic the direction of falling tempera- 
tures is not toward the invariant point on all three of the boundaries meeting at 
the point. As a result, some compositions complete their crystallization at the 
invariant point, while others do not. Three invariant points which are not eutec- 
tics appear in figure 9. At h and J 2 the direction of falling temperatures is away 
from the invariant point on one boundary. At 7 3 the direction of falling tempera- 
tures is away from the point on two boundaries. 

Crystallization after the liquid arrives at a non-eutectic invariant point may be 
illustrated in the case of I 2 . Mixtures in the triangle A-A n C-B complete then 
crystallization with the liquid at h . On the other hand, when some mixtures in 
the triangle A n C-AC-B are cooled, the liquid arrives at h at 62° and remains at 
that point, without change of temperature, until all of the solid A which may be 
present has disappeared. The liquid then follows the boundary I 2 -Ii to I h where 
crystallization is completed. 

Systems of maximum and minimum heat content 
From figure 9 it is learned that at 62° the liquid h may exist in equilibrium 
with the three solid phases A, A n C y and B. All mixtures of the four phases must 
lie in the quadrilateral formed by joining points representing the compositions of 
the phases, that is, the quadrilateral A-A n C-h-B (figure 10). No mixtures 
outside of this quadrilateral can be composed of the four phases,— liquid h and 
solids A, A n C y and B. 

Any three of the four phases may exist in equilibrium at 62°, giving the 
following systems of three phases: 

I. Solid A y solid A n C, and liquid I 2 

II. Solid A, solid B, and liquid h 

III. Solid A n C, solid £, and liquid J 2 

IV. Solid A, solid A n C, and solid B 

The diagonal A-h divides the quadrilateral into two triangles, A-A n C-U and 
A-B-h, designated as system I and system II, respectively, in figure 10. These 
are systems of maximum heat content at 62°, as may be known from the fact 
that the solid phases represented at their vertices are primary phases in regions 
separated by a boundary on which the direction of falling temperature is toward 
J*. For example, in system I the solid phases are A and A n C. The A and A n C 
primary-phase regions, figure 9, are separated by the boundary curve h-Iz, on 
which the temperature decreases as h is approached. Systems I and II are also 
systems of maximum liquid content at 62°. 

The diagonal A n C-B divides the quadrilateral A-A n C-h-B into two triangles, 
A n C-B-h and A-A n C-B, designated as systems III and IV, respectively, in figure 
11. These are systems of minimum heat content. Mixtures in system III are 



116 



L. A. DAHL 



composed of liquid h, solid A n C\ and solid B when equilibrium is attained at 
62°, with minimum heat content. That is, they are not crystallized completely 
at 62°. Upon cooling below 62°, with a continuous attainment of equilib- 
rium, the liquid phase changes in composition along the boundary curve Ir-Ii t 
the direction of falling temperatures being away from 7 2 toward I±. 

Any mixture in the quadrilateral A-A n C-I r B is in either system I or system 
II, figure 10, with maximum heat content at 62°, and is in either system III or 
system IV, figure 11, with minimum heat content at G2°. For example, a mix- 
ture of composition M (figures 9, 10, and 11) is in system II, figure 10, when it is 



! 




Proportion 
cf5oiid 3 



Proportion of Liquid lz 



Proportion of 5c fid A 



Fig. 10. Systems of maximum heat content at temperature at invariant point 
maximum proportion of Liquid at that temperatun 

at maximum heat content at 62°, and is composed of solid phases A and B and 
a maximum of liquid 7 2 - When it is at minimum heat content at 62 c it is in 
system IV, figure 11, and is composed of the three solid phases A, A n C\ and B. 

The phase composition of mixture M at maximum and minimum heat content 
is estimated in figure 10 and 11, respectively. It can be seen by comparing the 
two figures that the proportion of solid A decreases in passing from maximi 
minimum heat content. That is, there is more solid A present with a maximum 
amount of liquid h than when crystallization is complete. 

The decrease in quantity of solid A during crystallization at invariant point 
It is of particular interest. The lines required in figures 10 and 11 for estimating 



^ 




PtoporHon of Solid B \fhiporfon of Solid ArC Proportion of Solid A 



6 



Fig. 11. Systems of minimum heat content at temperature at invariant point. (Mini- 
mum of liquid h in system III, and complete crystallization in system IV.) 




Proportion ofSoi'd A, w/-rh 

complete cryshliizationi 
* *-i 



fhoporfion of Solid A wiih 

maximum liquid at 62° i 
* - *— * 



5 



Fig. 12. Diagram illustrating reduction in proportion of one solid phase with crystalliza- 
tion at invariant point not a eutectic. 

117 



I^MH 






118 L. A. DAHL 

the proportion of solid A in these figures have been transferred to figure 12. The 
line MS in figures 10 and 12 is parallel with the line I 2 B, The line MV in figures 
11 and 12 is parallel with the line A n C-B. The location of the invariant point 
Ii with reference to the triangle A-B-A n C determines whether the length of VB . 

will be greater or less than the length of SB, that is, whether the proportion of 
solid .1 will increase or decrease as crystallization proceeds at point J 2 . If it were < 

inside the triangle A-B-A n C y that is, if I 2 were a eutectic point, VB would be 
greater than SB, and solid A would increase in quantity during crystallization at 
I t . Since point I 2 is outside the triangle A-B-A H C, solid A decreases in quantity 
as composition M completes its crystallization at h. This would be true for any 
composition in the triangle A-B-A n C. In general, when an invariant point is 
outside of the triangle which includes all compositions completing their crystalli- 
zation at that point, in a direction away from a vertex X, the solid* phase X is 
present in greater quantity when the liquid arrives at the invariant point than 
when crystallization is completed. 

The decrease in solid A during crystallization at an invariant point is not a 
mere physical transfer of the molecular species A from one phase to another, since 
the other phases present when crystallization is completed are the pure solid 
phases B and A n C. The process of crystallization is not a purely physical proc- 
ess but involves chemical change. 

x role oftfie liquid phase in processes of fusion and crystdUUzaU 

In dealing with systen dch the number of primary-phase region- fa not 

greater than the number of components, as in figure 1 , the pr ad 

-tallization may be treated as purely physical processes. When then 
more primary-phase region- than the number of components, and the invariant 
■ re eutectics, as in figure 6, the course of crystallization, starting with the 
liqii may be treated as a physical p However, in considering the 

process of fu ■•mplete or partial, the chemical changes occurring in the 

liquid phase musl be recognized. In the case of systems in which tin n- 

eutectic invariant points, as in figure 9, the fact that chemical change- occur in 

n Is apparent. 

in processes involving partial fusion of refractory materials the chemical re- 

li produce the mineral constituents in the product occur in the liquid 

For example, portland cement raw mixta converted into the 

Jfi very slowly at temperatures below that of liquid formation. 

When the temperature is raised to a point at which liquid appears, chemical 

ir rapidly. The liquid not onh a medium of rapid exchange 

for the various components, hut its equilibrium relations with the various possible 

solid phases in the system determine the solid phases which will he form 

1 >( - ' 'the course of crystallization in reports of investigations of pi 

equilibria usually assume an ideal condition in which cooling is so slow 
equilibrium is attained at each temperature. Crystalliaatioi 
ical process, with departures from this treatment only when 

chemical change is evident, and cannot be ignored. This 



PHASE DIAGRAMS OF TERNARY SYSTEMS 119 

treatment is proper, since it is impossible to predict all of the possible applications 
of the phase equilibria data, and the conditions under which departures from the 
ideal conditions may become important, On the other hand, application of the 
phase diagram to industrial problems may require consideration of the manner in 
which equilibrium is attained in the process of fusion, and also the effects of 
failure to attain equilibrium during fusion or crystallization. In such applica- 
tions the role of the liquid phase must be taken into account. 



■■